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3443
Subject
Industrial Engineering
Date
Feb 20, 2024
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3
Uploaded by ConstableFish3557
EGN 3443.002: Probability and Statistics for Engineers
Quiz 1
Due: Sept. 9
st
(9:30PM), 2022
Problem 1 (1.5 points)
A sample of 24 offshore oil workers took part in a simulated escape exercise,
resulting in the accompanying data on time (sec) to complete the escape, as shown
in Table 1.
Table 1: A sample data for escape time
389
356
363
375
424
394
402
373
373
370
364
325
339
393
392
369
374
359
366
364
356
403
334
397
(a) Construct a stem-and-leaf display of the data, using the hundreds and tens
digits as the stem for each observation.
(0.3 point)
(b) Calculate the sample mean, sample median, and sample standard deviation.
(0.3 point)
(c) By how much could the largest time, currently 424, be increased without affect-
ing the value of the sample median? By how much could this value be decreased
without affecting the value of the sample median?
(0.3 point)
(d) Fill in the blanks of the following table.
Then, draw a histogram with the
relative frequency.
(0.3 point)
Bin
Frequency
Relative Frequency
[320,340)
[340, 360)
[360, 380)
[380, 400)
[400, 420)
[420, 440]
1
(e) Calculate the sample mean, sample median, and sample standard deviation
when the obsevations are re-expressed in minutes.
(0.3 point)
Note:
For parts (b) (c) and (e), to get full points, include intermediate steps.
Problem 2 (1.5 points)
Considering a sample of
n
observations,
x
1
,
x
2
,...,
x
n
, and letting
y
i
=
2
x
i
−
5
,
i
=
1
,...,
n
, answer the following questions.
(a) How do the sample mean ¯
y
of
y
i
’s relate to the sample mean ¯
x
of
x
i
’s? Derive
Verify your conjecture using analytical derivation.
(0.5 point)
(b) How do the sample median ˜
y
of
y
i
’s relate to the sample median ˜
x
of
x
i
’s?
Verify your conjecture using analytical derivation.
(0.5 point)
(c) How do the sample variance
s
2
y
of
y
i
’s relate to the sample median
s
2
x
of
x
i
’s?
Verify your conjecture using analytical derivation.
(0.5 point)
Note:
To get full points, include intermediate steps.
Problem 3 (1.5 points)
Consider randomly selecting a student at a large university. Let A be the event
that the selected student has a Visa card, let B be the event that the selected student
has a MasterCard and let C be the event that the selected student has an American
Express card. Suppose that
Pr
(
A
) =
0
.
6,
Pr
(
B
) =
0
.
4,
Pr
(
C
) =
0
.
2,
Pr
(
A
∩
B
) =
0
.
3,
Pr
(
A
∩
C
) =
0
.
15,
Pr
(
B
∩
C
) =
0
.
1 and
Pr
(
A
∩
B
∩
C
) =
0
.
08. Answer the
following questions.
(a) What is the probability that the selected student has at least one of the three
types of cards?.
(0.3 point)
(b) What is the probability that the selected student has both a Visa card and a
MasterCard but not an American Express card.
(0.3 point)
(c) Calculate and interpret
Pr
(
B
|
A
)
and also
Pr
(
A
|
B
)
.
(0.3 point)
(d) If we learn that the selected student has an American Express card, what is the
probability that she or he also has both a Visa card and a MasterCard?
(0.3 point)
(e) Given that the selected student has an American Express card, what is the
probability that she or he has at least one of the other two types of cards?
(0.3
point)
Note:
To get full points, include intermediate steps.
2
Problem 4 (1.5 points)
A manufacturing firm employs three analytical plans for the design and devel-
opment of a particular product.
For cost reasons, all three are used at varying
times. In fact, plans 1, 2, and 3 are used for 35%, 25%, and 40% of the products,
respectively. The probabilities of a defective product, given plans 1, 2, and 3, are
0.02, 0.01 and 0.03, respectively.
(a) What is the probability that a randomly selected product is defective?
(0.5
point)
(b) If a random product is observed and found to be defective, what is the proba-
bility that this product is the result of the use of plan 1?
(0.5 point)
(c) If a random product is observed and found to be defective, which plan is most
likely used and thus responsible?
(0.5 point)
Note:
To get full points, include intermediate steps.
3
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